EXAMPLES. 23141+31811+ 98365 +372 +19 = 1262 +3557 +4634 +12 +7376 +93172 = 1 = Proof of Addition.-Addition may be proved by beginning at the top of the column, and adding the figures in a contrary direction, or by cutting off by a line the upper number, and taking the sum of all the numbers below it, and then adding this sum to the upper number. If, in either case, the result agrees with the former, the work is presumed to be correct. SUBTRACTION. (8.) Subtraction consists in finding the difference between two numbers. The greater of the two numbers is called the minuend, the less the subtrahend, and the difference obtained, the remainder. The sign of subtraction is, and is read minus; 8-4 signifies, that 4 is to be subtracted from 8. RULE. Write the numbers, as in addition, placing the less number under the greater. Begin at the right, and take each figure of the subtrahend from the corresponding one of the minuend; but if the figure in the minu nd be the least, add ten to it, and subtract as before, always in such cases carrying one to the next figure of the subtrahend, to compensate for the ten added to the minuend. Demonstration of the lule.-To show the correctness of this rule, we remark that the difference of two numbers is evidently the difference of the units, tens, hundreds, &c. which compose these numbers; and hence, when the figures in the subtrahend are all less than those in the minuend, the sum of the differences of the several orders of units. will be the difference of the numbers. When the figure in the minuend is less than the corresponding one in the subtrahend, and the rule requires us to add ten to it ;-by doing this we but borrow one from the next higher order, which be comes ten by moving it one place to the right; the figure from which this was borrowed ́ought therefore to be diminished by one, or the figure below it increased by one, which evidently gives a like result. Proof. The proof of subtraction consists in adding together the remainder and the subtrahend; the sum must equal the minuend. EXAMPLES 1. From 9092 subtract 7835 9092 7835 1257 Having written the numbers as directed in the rule, I commence at the right, and since 5 cannot be taken from 2, I borrow 1 from the next place to the left. This, moved one place towards the right, becomes 10, which added to 2, gives 12, from which 5 can be subtracted. The remainder 7 is written below, and I proceed to the next place of figures to the left. Now since 1 was borrowed from 9, I may either consider it diminished by one, and take 3-from 8, or adding 1 to 3 take 4 from 9; either of which methods gives 5 for a remainder. Proceeding to the next figure, since 8 cannot be subtracted from 0, I add 10 to it: taking 8 from 10, 2 remains, and 1 is to be carried to 7, which gives 8 to be subtracted from 9, the remaining figure of the minuend. Writing the difference 1, below, I find the whole difference 1257, which added to the subtrahend, will produce the minuend. If the left hand figure in the minuend had been less than the figure immediately below it in the subtrahend, the subtraction would have been impossible, since there are no figures to the left from which to borrow. Indeed, in that case, the minuend would have been less than the subtrahend; and it is evident that the greater number cannot be taken from the less. 2. 322974 — 123748 = 199226 3. 1002641-783219 = 6. From 3 millions 31 thousands and 1, take 3721. 7. From 1 billion 100 millions 2 thousands and 2, take 810,032. 8. From 1 thousand 1 hundred and 11, take 112. MULTIPLICATION. (9.) Multiplication is the successive addition of a number to itself a given number of times. The sign of multiplication is x; 8×3 signifies that 8 is to be multiplied by 3, or repeated three times, and is equal to 8+8+8=24. The number to be repeated is called the multiplicand; the number of times it is to be repeated is called the multiplier; and the result is called the product. The multiplier and multiplicand are also called factors of the product. Any two numbers which multiplied together produce a third number, are called factors of that number. One number may therefore have many different factors; thus the number 72 may have for factors, 6 and 12, 8 and 9, 3 and 24, since 6x 12, 8 × 9, and 3 × 24, are each equal to 72. (10.) Before giving the rule, we will prove 1st, that the product of two numbers is the same in whatever order they are multiplied; and 2d, that where the multiplier can be divided into factors, the product can be obtained by multiplying by these factors successively. Let the numbers to be multiplied be 5 and 6. It is to be proved that 6 multiplied by 5 is the same as 5 multiplied by 6. Let the figure 1 be written five times in a horizontal line, and let six such lines be written: then considering 5 the multiplicand, we have it repeated six times, and the product equal to 30; but the number of units is the same, whether we consider the five units in the horizontal line repeated six times, or the six units in the vertical line repeated five times; but in this case we should have 6 multiplied by 5. We may make use of the same diagram to prove the second principle laid down. Let 5 be the multiplicand and 6 the multiplier, and suppose 6 divided into its factors 2 and 3. First we can represent the product of 5 by 3, by writing three lines with five units in each, and then repeat these three lines to represent the multiplication by 2, which gives 5 repeated six times. These same results would be obtained whatever numbers are used, and hence we are authorized to apply the principles laid down to all numbers. RULE I. When the multiplier is a single figure. (11.) Set the multiplier, under the right hand figure of the multiplicand, and, beginning at the right, multiply each figure of the multiplicand by the multiplier. Set down, in the product, the excess above the tens, and carry the tens as in addition, adding them to the product of the multiplier by the next figure of the multiplicand. EXAMPLE. *7835924678 8 62687397424 Demonstration.-To repeat a number a given number of times is, to repeat the units, tens, hundreds, &c. which it contains; but if we repeat eight units eight times, we shall have sixty four units or six tens and four units. The tens belong to a higher order of units, and therefore must be carried one place to the left, as in addition. The same reasoning will apply to any of the higher orders of units. RULE II. When the multiplier contains more than one figure. (12.) Multiply the multiplicand by each figure of the multiplier, placing the right hand figure of each product directly under that figure of the multiplier by which it is produced, and take the sum of all the products. NOTE. When there are O's in either of the factors, since the product of O by any number whatever, must be 0, it follows, that there can be no partial product to correspond to the place of 0 in the product, unless some number is carried to this place from the preceding product. When there are O's at the right of the multiplier, the multiplication commences at the first significant figure, and the O's must be written at the right of the product. Demonstration.-The demonstration of the preceding rule will best be understood by reference to an example. Suppose it required to multiply 4322 by 654. When we multiply the units of the multiplicand by 4, the units of the multiplier, the product will be units, and therefore stands in the unit's place: but when we multiply 2 by 5 tens, or, which is the same thing, (10) multiply 5 tens by 2, the product will be tens, and must therefore stand in the ten's place. For a like reason, the product of hundreds by the units, must be hundreds, and must be put in the place of hundreds; and in general, the product of the units by any figure of the multiplier will be of the same order as that figure, and is therefore to be placed directly beneath it in the product. 4322 654 17288 21610 25932 2826588 RULE III. When the multiplier can be divided into factors. (13.) Multiply first by one of the factors, and the resulting product by the other, the last will be the product required. The reason for this rule will be understood by a reference to Art. (10). |